4.2. Fuzzy regression discontinuity design and IV estimation
In the sharp RD design, treatment depends on the running variable s in a deterministic manner. However, in reality, treatment assignment is likely to depend on s in a stochastic manner, which is referred to in the literature as the fuzzy RD design. 15 In this case, OLS estimates of Eq. (4) may be biased. In our context, a man may retire before the age of 60 or continue to work after the age of 60. In this case, the OLS estimate of α1 in Eq. (4) using the variable R ispt could be subject to selection bias. To address this issue, we introduce the second treatment variable E ispt . E ispt is equal to 1 if the husband’s age s is above 60 but equal to 0 if s is below 60. The variable E ispt itself does not suffer from fuzziness and can be used to cleanly estimate an intent-to-treat effect. However, the impact of eligibility is not of primary interest; our goal is to estimate the impact of actually retiring on consumption. To obtain an unbiased estimate of this effect, we can use E ispt as an instrument for R ispt , because E ispt strongly predicts R ispt but is not subject to selection bias ( Imbens and Lemieux, 2008 ). One caveat is that the IV estimate is local average treatment estimate (LATE), meaning that the results can only be applied to households whose husbands comply with the mandatory retirement policy.